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What is the definition of a "parametric model"? And can you give a reference for the statement that the "extensional" quotient of the overall-initial model is the initial parametric model? On Wed, Aug 27, 2025 at 9:20 PM Andrew Polonsky <[email protected]> wrote: > [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list > ] > > Put > Top = \forall X. X -> X > tt = /\X.\x:X.x : Top > T(X) = (X -> Top) -> (Top -> Top) > A1 = A2 = Top > R \subset A1xA2 is =beta(eta) > T(R) identifies \x.x, \xy.y, and \xy.tt. > > While I agree that the term "initial model" is most often applied to a term > model with a syntactic notion of equality, in discussions of IEL there's a > general implicature that you are interested in parametric models. And the > initial such model is precisely the extensional quotient of the term model, > for which IEL holds tautologically. (More precisely, it holds by the > fundamental theorem of logical relations, the version for open terms that > is verified in the proof of the theorem.) > > Best, > Andrew > > > On Wed, Aug 27, 2025 at 7:59 PM Ryan Wisnesky <[email protected]> wrote: > > > Thanks everyone for the informative discussion! Andrew’s > counter-example > > of T(X) = X->X to the IEL in the initial model is just what was asked > for. > > > > If I may ask a follow-up: does anyone have a counter-example for the IEL > > in the initial model for a type expression that defines a functor? Or a > > proof that IEL always holds for definable functors in the initial model? > > (T(X) = X->X is well-known not to be a functor.). > > > > Thanks again! > > Ryan > > > > > On Aug 14, 2025, at 10:59 PM, Andrew Polonsky < > [email protected]> > > wrote: > > > > > > [ The Types Forum, > > http://lists.seas.upenn.edu/mailman/listinfo/types-list ] > > > > > > Yes, the initial model of System F satisfies the identity extension > > lemma, > > > assuming that by "initial model" you mean the extensional quotient of > > > closed terms as treated e.g. by Hasegawa [1]. > > > > > > The identity relation in this model is the extensional equality defined > > by > > > induction on type structure with the usual logical relation conditions. > > > This identifies terms that are not beta(-eta) convertible, like the two > > > successors on the polymorphic church numerals. > > > > > > Nat = \forall A. (A -> A) -> (A -> A) > > > S, S' : Nat -> Nat > > > S = \nfs.f(nfs) > > > S' = \nfs.nf(fs) > > > > > > If by "initial model" you mean something intensional like a lambda > > algebra > > > with beta conversion, then IEL fails. We can use the same type as the > > > counterexample. > > > > > > T(X) = X -> X > > > A1 = A2 = Nat > > > R \subset A1xA2 is =beta > > > T(R) identifies S and S' > > > > > > Best, > > > Andrew > > > > > > P.S. Regarding the other question raised in the Zulip chat you linked, > > > whether every type A is isomorphic to "forall X. (A -> X) -> X" in this > > > extensional model, I believe the answer to be "yes" based on comments > by > > > Thierry Coquand in [2], slide 52. > > > > > > [1] > > > https://urldefense.com/v3/__https://doi.org/10.1007/3-540-54415-1_61__;!!IBzWLUs!S8aJh356wvmDdcHzuNn_x-n_pAINEuB9OorQS4vpVVPxzpk6f1Frz6bXU2iKOmWU2eJnfiof4UvCQ7Pz7EY2e8iQYns02p8_gTB7LQ$ > > > [2] > > > https://urldefense.com/v3/__https://youtu.be/6Ao9zXwyteY?si=5FdC5t5Mnyajmchz&t=4660__;!!IBzWLUs!S8aJh356wvmDdcHzuNn_x-n_pAINEuB9OorQS4vpVVPxzpk6f1Frz6bXU2iKOmWU2eJnfiof4UvCQ7Pz7EY2e8iQYns02p8az4OKSg$ > > > > > > > > > On Tue, Aug 12, 2025 at 12:02 PM < > > [email protected]> > > > wrote: > > > > > >> Send Types-list mailing list submissions to > > >> [email protected] > > >> > > >> To subscribe or unsubscribe via the World Wide Web, visit > > >> https://LISTS.SEAS.UPENN.EDU/mailman/listinfo/types-list > > >> or, via email, send a message with subject or body 'help' to > > >> [email protected] > > >> > > >> You can reach the person managing the list at > > >> [email protected] > > >> > > >> When replying, please edit your Subject line so it is more specific > > >> than "Re: Contents of Types-list digest..." > > >> > > >> > > >> Today's Topics: > > >> > > >> 1. Identity extension for the system F term model (Ryan Wisnesky) > > >> > > >> > > >> ---------------------------------------------------------------------- > > >> > > >> Message: 1 > > >> Date: Mon, 11 Aug 2025 21:25:19 -0700 > > >> From: Ryan Wisnesky <[email protected]> > > >> To: "[email protected]" > > >> <[email protected]> > > >> Subject: [TYPES] Identity extension for the system F term model > > >> Message-ID: <[email protected]> > > >> Content-Type: text/plain; charset=utf-8 > > >> > > >> Hi All, > > >> > > >> I'm hoping the folks on this list can settle a question of folklore; > > >> myself and Mike Shulman and others have been discussing it on the > > applied > > >> category theory zulip channel but have yet to reach a conclusion: > > >> > > > https://urldefense.com/v3/__https://categorytheory.zulipchat.com/*narrow/channel/229199-learning.3A-questions/topic/Continuations.2C.20parametricity.2C.20and.20polymorphism/with/528479188__;Iw!!IBzWLUs!Uf89Hb7uWjtbz4qUeFbMCVwfxkuye2lpB27Oi4vChhMV6yMPpA57jl2oZnBt3TaOOYa7UhjQTz16dPMUZoGtGubQesXT$ > > >> > > >> The question is whether the initial (term) model of system F (2nd > order > > >> impredicative polymorphic lambda calculus) satisfies the "identity > > >> extension lemma", which is one of the primary lemmas characterizing > > >> (Reynolds) parametric models. To be clear, this question is about the > > >> initial (term) model of system of F, its initial model of contexts and > > >> substitutions. > > >> > > >> There are many places that state the system F term model should obey > > >> identity extension, for example, this remark by Andy Kovacs: > > >> > > > https://urldefense.com/v3/__https://cs.stackexchange.com/questions/136359/rigorous-proof-that-parametric-polymorphism-implies-naturality-using-parametrici/136373*136373__;Iw!!IBzWLUs!Uf89Hb7uWjtbz4qUeFbMCVwfxkuye2lpB27Oi4vChhMV6yMPpA57jl2oZnBt3TaOOYa7UhjQTz16dPMUZoGtGpQSw3Ii$ > > >> . > > >> > > >> However, neither myself nor Mike nor anyone on the zulip chat or > anyone > > >> we?ve asked has been able to find a proof (or disproof). > > >> > > >> Anyway, do please let me know if you know of a clear proof or disproof > > of > > >> the identity extension lemma for the initial (term) model of system F! > > >> > > >> Thanks, > > >> Ryan Wisnesky > > >> > > >> PS Here's a statement from Mike about the exact definition of this > > >> question: > > >> > > >> Let C be the initial model of system F. Let R(C) be the relational > > model > > >> built from C, so its objects are objects of C equipped with a binary > > >> relation. There is a strict projection functor R(C) -> C, and since C > > is > > >> initial this projection has a section C -> R(C), which is "external > > >> parametricity". In a theory like System F that has type variables, a > > >> "model" includes information about types in a context of type > > variables, so > > >> external parametricity sends every type in context to a "relation in > > >> context". Do the relations-in-context resulting in this way from > types > > of > > >> System F always map identity relations to identity relations? > > >> > > >> ------------------------------ > > >> > > >> Subject: Digest Footer > > >> > > >> _______________________________________________ > > >> Types-list mailing list > > >> [email protected] > > >> https://LISTS.SEAS.UPENN.EDU/mailman/listinfo/types-list > > >> > > >> > > >> ------------------------------ > > >> > > >> End of Types-list Digest, Vol 155, Issue 3 > > >> ****************************************** > > >> > > > > >
